Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 4 / 25J / i / Solution

Work in the real $L^1$ space, consistent with the specified real step coefficients. First approximate an integrable function by a finite-valued measurable function with bounded support. Truncate its domain to $[-M,M]$ and its values to $[-M,M]$; the $L^1$ error tends to zero by integrability. Quantize those remaining values in intervals of mesh $\delta$, producing a simple function $u={\sum }_{j=1}^m{c}_j{1}_{B_j}$ with finite-measure Borel sets $B_j$ and error at most $2M\delta$ on the truncated domain. Thus $\Vert f-u{\Vert }_1$ can be made arbitrarily small.

For each $B_j$, use the allowed approximation by a finite union of intervals $A_j$, choosing the symmetric-difference errors soThese unions can be written as finite disjoint bounded intervals, and changing endpoints to the required half-open convention changes only a null set. The function $s={\sum }_j{c}_j{1}_{A_j}$ is therefore a step function of the specified form, andCombining the two approximations proves the step functions are dense in $L^1(\mathbb{R})$. In the complex version of $L^1$, the same argument applies to real and imaginary parts when complex step coefficients are allowed.